Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 320

Proposition 9.2 ([ 29 ]). If the optimal anti-angiogenic dose rate u and the
radiotherapy dose rate w both follow singular regimens u sin and w sin on an open
interval I, then, in addition to Eq. ( 43 ), a second relation of the form

B ( x ( t )) u sin ( t )+ w sin ( t )= A ( x ( t )) (44)

holds on I where A and B are smooth functions that only depend on the dynamics of
the system and can be determined analytically.

2 system of linear equations
whose coefficients are determined solely by the equations defining the dynamics of
the system. It is possible to give explicit expressions for the functions A and B and
thus also for the controls. However, these formulas depend on the second derivatives
of the terms in the dynamics and they are long and unwieldy. On the other hand,
given any particular value

Overall, ( u sin , w sin ) thus are the solutions of a 2 ×

of the state and specified values of the parameters,
it is not difficult to compute these coefficients A and B numerically and solve for the
controls.
Figure 3 gives an example of a totally singular anti-angiogenic dose rate u and a
radiotherapy dose rate w that have been computed in this way for parameter values
taken from [ 8 ]. Part (a) shows the graph of the radiation schedule if no upper limit
on the dose rate is imposed. If we set the radiation limit to w max =

( p , q )

5, then this upper
bound is initially exceeded and part (b) shows the control that has been computed
by saturating this schedule at w max .SinceEq.( 43 ) is valid regardless of the structure
of w , the calculations easily adjust. The corresponding graph of the singular control
u is given in part (c). Part (d) shows the corresponding trajectory. Note that this
trajectory is almost linear. In fact, whenever the anti-angiogenic control u follows a
singular regimen, then the quotient

p
q

follows the simple dynamics

p
q

d
d t

2
3 ξ

d

b p 3

= (45)

. .

and, along this simulation, the right-hand side only varies between 0 03 and 0 09,

i.e., is almost constant.
The controls given in this figure were not computed to be optimal, but they
only illustrate totally singular controls for a combination of anti-angiogenic and
radiotherapy. Based on our theoretical analysis, it is clear that these controls will
play an essential part in the structure of optimal protocols. This is seconded by
the structure of optimal protocols computed in [ 8 ] where all the solutions given
have exactly this structure, but no hard limits on the dosage rates were imposed.
In order to solve the overall optimal control problem, however, it is necessary to
take these constraints into account and then to establish the structure of optimal
controls before and after the singular segments. Different from the monotherapy
problem described earlier, in this case there exists a vector field whose integral
curves are the trajectories for totally singular controls everywhere, not just on some
lower-dimensional surface. However, it matters which of these trajectories is taken.
Research on determining an optimal synthesis is ongoing.